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Theorem suplocexprlemlub 7725

Description: Lemma for suplocexpr 7726. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

**Hypotheses**
Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<_P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<_P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<_P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<_P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩

**Assertion**
Ref	Expression
suplocexprlemlub	⊢ (𝜑 → (𝑦<_P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<_P 𝑧))

Distinct variable groups: 𝑦,𝐴,𝑧 𝑥,𝐴,𝑦 𝑧,𝐵 𝜑,𝑦,𝑧 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑤,𝑢) 𝐴(𝑤,𝑢) 𝐵(𝑥,𝑦,𝑤,𝑢)

Proof of Theorem suplocexprlemlub Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑦<_P 𝐵) → 𝑦<_P 𝐵)
2		ltrelpr 7506	. . . . . . . 8 ⊢ <_P ⊆ (P × P)
3	2	brel 4680	. . . . . . 7 ⊢ (𝑦<_P 𝐵 → (𝑦 ∈ P ∧ 𝐵 ∈ P))
4	3	simpld 112	. . . . . 6 ⊢ (𝑦<_P 𝐵 → 𝑦 ∈ P)
5	4	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<_P 𝐵) → 𝑦 ∈ P)
6	3	simprd 114	. . . . . 6 ⊢ (𝑦<_P 𝐵 → 𝐵 ∈ P)
7	6	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<_P 𝐵) → 𝐵 ∈ P)
8		ltdfpr 7507	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝐵 ∈ P) → (𝑦<_P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵))))
9	5, 7, 8	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑦<_P 𝐵) → (𝑦<_P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵))))
10	1, 9	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑦<_P 𝐵) → ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))
11		simprrr 540	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → 𝑠 ∈ (1^st ‘𝐵))
12		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩
13	12	fveq2i 5520	. . . . . . . . 9 ⊢ (1^st ‘𝐵) = (1^st ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩)
14		npex 7474	. . . . . . . . . . . . 13 ⊢ P ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → P ∈ V)
16		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
17		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<_P 𝑥)
18		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<_P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<_P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<_P 𝑦)))
19	16, 17, 18	suplocexprlemss 7716	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
20	15, 19	ssexd 4145	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ V)
21		fo1st 6160	. . . . . . . . . . . . 13 ⊢ 1^st :V–onto→V
22		fofun 5441	. . . . . . . . . . . . 13 ⊢ (1^st :V–onto→V → Fun 1^st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1^st
24		funimaexg 5302	. . . . . . . . . . . 12 ⊢ ((Fun 1^st ∧ 𝐴 ∈ V) → (1^st “ 𝐴) ∈ V)
25	23, 24	mpan 424	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (1^st “ 𝐴) ∈ V)
26		uniexg 4441	. . . . . . . . . . 11 ⊢ ((1^st “ 𝐴) ∈ V → ∪ (1^st “ 𝐴) ∈ V)
27	20, 25, 26	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (1^st “ 𝐴) ∈ V)
28		nqex 7364	. . . . . . . . . . 11 ⊢ Q ∈ V
29	28	rabex 4149	. . . . . . . . . 10 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V
30		op1stg 6153	. . . . . . . . . 10 ⊢ ((∪ (1^st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ∈ V) → (1^st ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩) = ∪ (1^st “ 𝐴))
31	27, 29, 30	sylancl 413	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩) = ∪ (1^st “ 𝐴))
32	13, 31	eqtrid 2222	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐵) = ∪ (1^st “ 𝐴))
33	32	eleq2d 2247	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (1^st ‘𝐵) ↔ 𝑠 ∈ ∪ (1^st “ 𝐴)))
34	33	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → (𝑠 ∈ (1^st ‘𝐵) ↔ 𝑠 ∈ ∪ (1^st “ 𝐴)))
35	11, 34	mpbid 147	. . . . 5 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → 𝑠 ∈ ∪ (1^st “ 𝐴))
36		suplocexprlemell 7714	. . . . 5 ⊢ (𝑠 ∈ ∪ (1^st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑠 ∈ (1^st ‘𝑧))
37	35, 36	sylib 122	. . . 4 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑠 ∈ (1^st ‘𝑧))
38		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → 𝑠 ∈ Q)
39	38	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑠 ∈ Q)
40		simprrl 539	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → 𝑠 ∈ (2^nd ‘𝑦))
41	40	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑠 ∈ (2^nd ‘𝑦))
42		simpr 110	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑠 ∈ (1^st ‘𝑧))
43		rspe 2526	. . . . . . . 8 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝑧))) → ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝑧)))
44	39, 41, 42, 43	syl12anc 1236	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝑧)))
45	4	ad4antlr 495	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑦 ∈ P)
46	19	ad4antr 494	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝐴 ⊆ P)
47		simplr 528	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑧 ∈ 𝐴)
48	46, 47	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑧 ∈ P)
49		ltdfpr 7507	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<_P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝑧))))
50	45, 48, 49	syl2anc 411	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → (𝑦<_P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝑧))))
51	44, 50	mpbird 167	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1^st ‘𝑧)) → 𝑦<_P 𝑧)
52	51	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) → (𝑠 ∈ (1^st ‘𝑧) → 𝑦<_P 𝑧))
53	52	reximdva 2579	. . . 4 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → (∃𝑧 ∈ 𝐴 𝑠 ∈ (1^st ‘𝑧) → ∃𝑧 ∈ 𝐴 𝑦<_P 𝑧))
54	37, 53	mpd 13	. . 3 ⊢ (((𝜑 ∧ 𝑦<_P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑠 ∈ (1^st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑦<_P 𝑧)
55	10, 54	rexlimddv 2599	. 2 ⊢ ((𝜑 ∧ 𝑦<_P 𝐵) → ∃𝑧 ∈ 𝐴 𝑦<_P 𝑧)
56	55	ex 115	1 ⊢ (𝜑 → (𝑦<_P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<_P 𝑧))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 {crab 2459 Vcvv 2739 ⊆ wss 3131 ⟨cop 3597 ∪ cuni 3811 ∩ cint 3846 class class class wbr 4005 “ cima 4631 Fun wfun 5212 –onto→wfo 5216 ‘cfv 5218 1^st c1st 6141 2^nd c2nd 6142 Qcnq 7281 <_Q cltq 7286 Pcnp 7292 <_P cltp 7296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-1st 6143 df-qs 6543 df-ni 7305 df-nqqs 7349 df-inp 7467 df-iltp 7471

This theorem is referenced by: suplocexpr 7726

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